In the closed-system setting of Ref. [1]}
the construction has been completed
by
the quasi-Hermiticity
\(Y_1^{\ddagger (2)}\,Z_2=Z_2\,Y_1\) 
holding in
\({\cal R}_2={\cal H}_{math}\) .
This relation
proves equivalent
to the
parity-charge pseudo-Hermiticity
of the admissible Hamiltonians.
In this sense, the explicit reference to
the auxiliary space
\({\cal R}_1\)  and even to
the physical Hilbert space \({\cal R}_0\)  itself becomes,
from the practical user's point of view,
redundant.
<TABLE>